00:01
For the problem three, we have the equation where the function s is a function of t, which is given over here.
00:06
We need to find its domain, which is pretty much straightforward because that's a polynomial, that's a polynomial, which means that the domain is all real values of x.
00:17
And we need to find where the function is increasing.
00:20
So that's where s prime comes in, because s prime t will give us an idea about where the function is increasing or decreasing.
00:27
So the differentiation of negative 16, negative 16 comes out and the differentiation of t squared is just 2t.
00:35
And for 1148t, that 114 .8 comes out and the differentiation of t is just one and the differentiation of 46 is 0.
00:43
So this is minus 32t plus 1148.
00:47
This is s prime t.
00:50
So when s prime t is equated, is positive, that will give us the values where, if is increasing the function is increasing so for part b we have negative 32 t plus 11148 should be greater than 0 if we if we multiply in fact let's divide both sides by in fact rather than dividing i'm going to take uh okay uh i'm going to take negative 32 as a common factor so i'm left with t minus 1148 over 32 and this is positive uh dividing both sides by negative 32 we get rid of this negative 32 and but make sure that since you're dividing with a negative number the inequality sign should reverse and this means that well let's see what is a common factor which with which we can reduce this i think four four times eight is 32 four times one one one four four is two eight so this is two hundred and eighty seven so this is t is less than two hundred and eighty seven or eight so this means that t should be less than two hundred and eighty seven over 8.
01:59
So this is the value of t where it is increasing and evidently where it is decreasing will be exactly the remaining part...